Method and apparatus for measuring polarization

ABSTRACT

An improved method and apparatus for the measurement of the polarization of light uses nonlinear polarimetry. The higher order moments of the E field are measured and then transformed into standard polarimetry parameters yielding the polarization of the light. In a first embodiment, the light to be measured is transmitted through a rotating retarder. The retarder is optically coupled to a fixed analyzer and light is then detected by linear and nonlinear photodetectors. The spectra from the detectors is calculated and transformed, to obtain the polarization. In a second embodiment, the light to be measured is received by an optical fiber comprising a plurality of fiber birefringences to retard the light. Polarization sensitive gratings along the length of the fiber scatter the light, and photodetectors detect the scattered light. Apparatus in two preferred embodiments can perform the inventive method.

FIELD OF THE INVENTION

[0001] This invention relates to a method and apparatus for measuringthe polarization of light.

BACKGROUND OF THE INVENTION

[0002] High-speed optical fiber communication systems operate byencoding information (data) onto lightwaves that typically propagatealong optical fiber paths. Most systems, especially those used formedium to long distance transmission employ single mode fiber. Asimplied by the name, single mode fibers propagate only one mode of lightbelow cutoff. The single mode typically includes many communicationschannels. The communications channels are combined into the onetransmitted mode, as by wavelength division multiplexing (WDM) or densewavelength division multiplexing (DWDM).

[0003] While only one mode is transmitted, that mode actually comprisestwo perpendicular (orthogonal) polarizations. The polarization of thesetwo components varies undesirably as the waves propagate along a fibertransmission path. The distortion of the optical signals caused by thevarying polarization is called polarization mode dispersion (PMD). PMDcan be corrected through a combination of measurements of the PMD andthe control of active corrective optics.

[0004] Polarimeters measure the polarization of light. Polarimeters cangenerate signals representing a measured degree of polarization that canbe useful for diagnostic purposes. The signals can also beadvantageously used for polarization correction using feedbacktechniques to minimize PMD.

[0005] Polarimeters generally employ one or more photodetectors andrelated electro-optical components to derive basic polarization data.The raw photodetector signal measurements are typically transformed bymathematical techniques into standard polarization parameters. In theprior art, the photodetector outputs are generally averaged, as by someelectronic time constant, and then multiplied as part of the signalprocessing and transformation process. The problem with averaging atdetection is that instantaneous temporal information lost throughaveraging cannot be retrieved later.

[0006] What is needed for more accurate polarization measurements is apolarimeter that instantaneously measures polarimeter photodetectoroutputs without averaging, multiplies the unaveraged signals early insignal processing, and then averages and transforms the signals intopolarimetry parameters.

SUMMARY OF THE INVENTION

[0007] An improved method and apparatus for the measurement of thepolarization of light uses nonlinear polarimetry. The higher ordermoments of the E field are measured and then transformed into standardpolarimetry parameters yielding the polarization of the light. In afirst embodiment, the light to be measured is transmitted through arotating retarder capable of rotating at a plurality of angles with atleast two retardances Δ. The retarder is optically coupled to a fixedanalyzer. The light from the analyzer is then detected by linear andnonlinear photodetectors. The spectra from the detectors is calculatedand transformed, to obtain the polarization. In a second embodiment, thelight to be measured is received by an optical fiber comprising aplurality of fiber birefringences to retard the light. Polarizationsensitive gratings along the length of the fiber scatter the light, andphotodetectors detect the scattered light. The signals from thephotodetectors can then be transformed to obtain the polarization.

[0008] Apparatus in two preferred embodiments can perform the inventivemethod. In the first embodiment, nonlinear and linear photodetectors arepreceded by a rotating retarder, rotating at a plurality of angles witha retardance, and an analyzer, such as a fixed polarizer. In the secondpreferred apparatus, a plurality of photodetectors are located adjacentto polarization sensitive gratings situated in a birefringent opticalwaveguide located between each of the polarization sensitive gratings.

DRAWINGS

[0009] The advantages, nature and various additional features of theinvention will appear more fully upon consideration of the illustrativeembodiments now to be described in detail in connection with theaccompanying drawings. In the drawings:

[0010]FIG. 1 shows an apparatus to perform the method of nonlinearpolarimetry;

[0011]FIG. 2 shows an alternative apparatus to perform the method ofnonlinear polarimetry using optical fiber and polarization sensitivegratings;

[0012]FIG. 3 shows an apparatus as in FIG. 1, including a linear and anonlinear detector;

[0013]FIG. 4 shows a detector arrangement comprising a linear and anonlinear detector;

[0014]FIG. 5 shows a first preferred embodiment of an apparatus toperform the method of nonlinear polarimetry; and

[0015]FIG. 6 shows a second preferred optical fiber apparatus to performthe method of nonlinear polarimetry.

[0016] It is to be understood that the drawings are for the purpose ofillustrating the concepts of the invention, and except for the graphs,are not to scale.

DETAILED DESCRIPTION

[0017] This description is divided into two parts. Part I describes theinventive method for polarimetry and two embodiments for makingpolarization measurements according to the inventive method. For thoseskilled in the art, Part II further develops, defines, and introducesthe concepts of invariance, state of polarization and degree ofpolarization, and the foundation equations governing nonlinearpolarimetry as best understood by applicants at the time of theinvention.

[0018] Part I: Nonlinear Polarimetry

[0019] Standard polarimeters use linear detectors and thus measure termsquadratic in the E field. These can be considered 2^(nd) order momentsof the E field and are related to the power and the Stokes parameter ofthe E-field. A detector measuring intensity squared, though wouldmeasure 4^(th) order moments of the E field. Such higher order momentscan have more information about the E-field. Simply put, a higher ordermoment of some time varying quantity is simply the time average of ahigher power of the quantity. The first power is always just the mean.The second power is the standard deviation and so on.

[0020] Nine moments of the E field can be measured with apparatus 10 asshown in FIG. 1. Incoming light 11 is retarded by retarder 12 at angleC, with a retardance of Δ. Analyzer A 13 is a polarizer that precedesthe nonlinear detector as represented by photodiode 14. This apparatuscan be accomplished using bulk optics or integrated electro-opticaltechniques.

[0021] Retarder 12 is an optical component that retards one polarizationwith respect to the orthogonal polarization. In terms of E fields, theretarder gives one of the polarizations a phase with respect to theother orthogonal E component. Examples are ½ λ or ¼ λ retarders.

[0022] A ¼ λ wave retarders causes a $\frac{\pi}{2}$

[0023] delay difference:$\Delta = {{\left( \frac{\frac{1}{4}\lambda}{\lambda} \right)2\quad \pi} = \frac{\pi}{2}}$

[0024] Similarly, a ½ λ wave retarders causes a delay difference of π:$\Delta = {{\left( \frac{\frac{1}{2}\lambda}{\lambda} \right)2\quad \pi} = \pi}$

[0025] Here, the retarder 12 is a generic retarder. It has an arbitraryangle and arbitrary phase retardance. The angle C sets the two linearstates of polarization on which the phase difference Δ is applied.

[0026] Analyzer 13 is a polarizer. It passes the light of polarizationA, and suppresses all other polarizations. By rotating analyzer 13,light of polarization A is a continuous sampling of all 2πpolarizations. A detector viewing the light output of a continuouslyrotating analyzer registers a periodic waveform. The Fourier spectra ofthat waveform contains a DC component (near 0), and all other componentsof the spectra.

[0027] A preferred alternative version of this embodiment rotatesretarder 12, with a fixed analyzer 13 to generate the sin and cosinequadrature components of the Fourier spectra of the detector output.These components yield the nine E field higher order components.

[0028] The response of the nonlinear detector is:V_(detector) = I_(optical)²   = [E_(x)² + E_(y)²]²

[0029] A linear detector would measure:$I = {\frac{1}{2}\left\lbrack {S_{0} + {\left\lbrack {{S_{1}\cos \quad 2C} + {S_{2}\sin \quad 2C}} \right\rbrack \cos \quad 2\left( {A - C} \right)} + \quad {\left\lbrack {{S_{2}\cos \quad 2C} - {S_{1}\sin \quad 2C}} \right\rbrack \sin \quad 2\left( {A - C} \right)\cos \quad \Delta} + \quad {S_{3}\sin \quad 2\left( {A - C} \right)\sin \quad \Delta}} \right\rbrack}$

[0030] The nonlinear detector would measure I², and the filter in the DCelectronics would determine an averaging time, as in the linear case:V_(detector) = ∫_(T_(RC  ))    t  I²=  …  ⟨S₀S₁⟩_(T_(R  C))  …

[0031] All nine components can be measured if one rotates both analyzerA and retarder C in a manner analogous to the linear Stokes case. Byperforming measurements at the different sum and difference frequenciesproportional to nine linearly independent superpositions of

S_(i)S_(j)

, a 9×9 inversion matrix may then be applied to calculate the ninemoments.

[0032] For the nonlinear polarimeter one would toggle between:${A = {\frac{\pi}{4} + ɛ_{1}}},{\Delta = {\frac{\pi}{4} + ɛ_{2}}},{{where}\quad ɛ_{1}\quad {and}\quad ɛ_{2}\quad {are}\quad {{small}.{and}}}$$A = {{\frac{\pi}{4}\quad {and}\quad \Delta} = \frac{\pi}{4}}$

[0033] and rotate C at a fixed rate. Then the nine moments can beextracted from the nine (quadrature) components: 1 (DC), cos 2C, sin 2C,cos 4C, sin 4C, cos 6C, cos 8C, sin 8C. However, it can be advantageousto have more oscillating components, since it is less desirable tomeasure at a frequency that appears in the DC or non-oscillatingresponse as this would be subject to DC noise.

[0034] A static measurement of the moments can also be done withapparatus 20 as shown in FIG. 2, but with polarization sensitivegratings 22 and fiber birefringences 23 for the retarder. Here nonlineardetectors 24 detect the light scattered by polarization sensitivegratings 22. Birefringent optical fiber 23 causes the birefringences. Inthe limit of weak scattering for each grating, the scattered E-field isthe same as in the case of the retarder and the analyzer. As before,there are nine detectors and a resultant 9×9 matrix to connect thedetector values to the moments.

[0035] Here each grating with its nonlinear detector 24 will generate anoutput signal which is proportional to a linear transformation of theStokes parameters. Each detector 24 signal is linearly related to aStokes tensor component. Therefore with proper grating 22 alignments,the nine detector 24 outputs have a linear relationship with the nineStokes tensor components. Gratings 22 are each aligned in differentdirections. Gratings 22 are each aligned azimuthally about the axis ofthe optical fiber. Both the grating 22 alignments and birefringences arealigned such that the 9×9 calibration matrix is invertible.

[0036] The measured moments have several uses. The degree ofpolarization (DOP) is most useful with the Stokes vector because it doesnot depend on the SOP. That is you can bump the fiber, and the DOP willnot change. In other words, the DOP is invariant (see definition ofinvariant later in Part II) under unitary or lossless transformations.This makes it valuable as a monitoring quantity since a fiber bump doesnot change it, at least not as much as a bump causes a change in S₁ orS₂. The higher order moments also have invariants. To understand theinvariance of

S_(i)S_(j)

, remember that

S₁S₂S₃

is a vector and unitary transformations correspond to a rotation on theStokes sphere R_(ij). With the higher order moments then, (

S₀S₁

,

S₀S₂

,

S₀S₃

) transforms as a vector. Therefore their magnitudes are fixed and:$\sum\limits_{1}^{3}{{\langle{S_{0}S_{i}}\rangle}{\langle{S_{0}S_{i}}\rangle}}$

[0037] is invariant.

[0038] But, there are more terms, since

S_(i)S_(j)

=T_(ij) is a tensor$\sum\limits_{i = 1}^{3}\quad {\sum\limits_{j = 1}^{3}\quad {T_{i\quad j}T_{i\quad j}}}$

[0039] is also invariant.

[0040] A proof of this is shown as follows (All duplicate indices aresummed from 1 to 3):

[0041] Rotation of the stokes tensor:T_(i  j)^(′)  R_(i  m)  R_(j  n) = T_(m  n),

[0042] invariant:T_(ij)T_(ij) = T_(ij)^(′)R_(im)R_(jn)T_(kl)^(′)R_(k  m)R_(i  n)   = T_(ij)^(′)T_(kl)^(′)R_(i  m)R_(mk)⁻¹R_(jn)R_(nl)⁻¹   = R_(mk)⁻¹R_(nl)⁻¹(these  are  3 × 3  rotation  matrices)   = ɛ_(ij)T_(ij)^(′)T_(ij)^(′)

[0043] Another invariant is:${\sum\limits_{1}^{3}\quad {T_{ii}T_{ii}}},$

[0044] and one can also get invariants from the determinants:

[0045] det(T_(ij)), where i,j=1,2,3. Therefore a list of some invariantsis:${\sum\limits_{i,{j = 1}}^{3}\quad T_{ij}^{2}},{\sum\limits_{j = 1}^{3}\quad T_{0j}^{2}},{\sum\limits_{j = 1}^{3}\quad T_{jj}^{2}},$

[0046] and det(T_(ij)), where i,j=1,2,3. These invariants can allrepresent useful monitoring quantities. Since higher moments are usuallymost interesting when combined with the lower moments to givefluctuations of the E-field, it would be useful to build in the samelinear measurement done in normal polarimetry.

[0047]FIG. 3 shows an apparatus to accomplish this measurementcomprising incoming light 31 retarded by retarder 32 at angle C, withretardance Δ. Analyzer A 33 comprises coupler 36, and nonlinear andlinear photon detector 34 and 35. The response of detectors isV_(l)=k_(l)I for detector 34, and V_(l)=k_(l)I² for detector 35. Bybuilding four more gratings into the device of FIG. 3, for a total of 13gratings, the averages can be subtracted from higher order moments.

[0048] Using such an embodiment, one can measure aV_(l) ²−V_(l), where αis such that when the signal is constant, aV_(l) ²−V_(n)=0, thenV_(n)−aV_(l) ²≧0, since intensity fluctuations always make

I²

>

I

². DOP=0 gives the extreme case, since the linear detector is constantin this case.

[0049] An important advantage to having both a linear and nonlineardetector is that the nonlinear detector can be “nonlinearized” bysubtracting out the linear part. This is illustrated by FIG. 4, wherethe response to light 41 of detector 34 is V_(l)=CI, and the response ofdetector 35 is V_(n)=aI²+bI. Thus:$V_{\underset{quadratic}{nonlinear}} = {{cV}_{n} - {{bV}_{i}{\alpha\alpha c}\quad I^{2}}}$

[0050] This would allow for lower powers to be used with the nonlineardetector. Of course the noise would still be as large as it is for onedetector, but one could extend the nonlinear concept previouslydiscussed and measure the linear and nonlinear moments simultaneously.This embodiment of the invention needs nine nonlinear (quadratic) andfour linear detectors. The 13 detectors would have a linear relationshipto the 13 linear and quadratic moments as related by a 13×13 matrix.Rotating polarizers or static birefringence can be used.

EXAMPLES

[0051]FIG. 5 shows a first preferred embodiment of the nonlinearpolarimeter. Here, rotating retarder 52 receives light 31. fixedpolarizer 53 is optically coupled to rotating retarder 52 and coupler36. Coupler 36 splits the light from fixed polarizer 53 to the twophotodetectors, linear detector 34 and nonlinear detector 35. Thisembodiment can be accomplished in bulk optics or by using integratedelectro-optics fabrication techniques.

[0052]FIG. 6 shows a second preferred embodiment of a nonlinearpolarimeter to accomplish static measurement of the moments. Here, fiber23 receives light 31. The light from polarization sensitive gratings 22is detected by four linear detectors 34 and nine nonlinear detectors 35.Each polarization sensitive gratings 22 has a different scatteringangle. Birefringent optical fiber 23 causes the birefringences. In thelimit of weak scattering for each grating, the scattered E-field is thesame as in the case of specific retarder and the analyzer positions. Asbefore, there are nine detectors and a resultant 9×9 matrix to connectthe detector values to the moments. This embodiment can be fabricatedwith optical fibers and fiber components or by integrated electro-opticfabrication techniques. Here the additional four detector outputs yielda 13×13 calibration matrix. The polarization sensitive gratings' 22scattering angles and the sections of birefringent optical fiber 23 areset such that the 13×13 calibration matrix is invertible.

[0053] Actual fabrication forms and techniques suitable for constructingthe inventive apparatus in general, includes, but is not limited to,bulk optical components, optical fibers and optical fiber components,and integrated techniques, including planer waveguides, and otherintegrated optical components.

[0054] Part II: Theoretical Development of Nonlinear PolarimetryIncluding the Definition of Invariance

[0055] Invariance: A polarization transformation is said to be invariantwhen there is a polarization transformation in which the two principlestates are delayed by less than the coherence length of the light. Thisis an invariant transformation. In mathematical terms:

[0056] ∫dtE₁(t)E₂(t+τ_(c))≠0, τ_(c)=correlation time, E₁,E₂ areprincipal states, and τ_(invariant)<<τ_(c). In short: Invariant=unitarywith τ<τ_(c) where τ is the maximum time delay between polarizationcomponents. Also the ratio of the two principle states must remainfixed, i.e., the “fiber touch” cannot be before a large PMD element suchas a fiber link, since changing the launch polarization into a fiberwith PMD will change the ratio of the two principle states and hencealter the output pulse shape and its higher order moments. The “fibertouch” that we wish to avoid being sensitive to through the use ofinvariants is that directly before the polarization monitor. Withstandard polarimeters the only invariants are the total power and theDOP.

[0057] State of polarization and Degree of Polarization: It is useful toprovide a clear definition of “state of polarization” (or SOP), withrespect to an optical signal propagating through a fiber. In general, ifthe core-cladding index difference in a given optical fiber issufficiently small, then the transverse dependence of the electric fieldassociated with a particular mode in the fiber may be written as:

E(z,t)={circumflex over (x)}A _(x)exp(iφ _(x))+ŷA _(y)exp(iφ _(y))

[0058] where A_(x) and A_(y) define the relative magnitude of eachvector component and the phases are defined as follows:

φ_(x) =βz−ωt+φ ₀, and

φ_(y) =βz−ωt+φ ₀−δ,

[0059] where β defines the propagation constant, ω defines the angularfrequency, φ₀ defines an arbitrary phase value, and δ is the relativephase difference between the two orthogonal components of the electricfield.

[0060] In accordance with the teachings of the present invention, thestate of polarization (SOP) of an optical fiber will be described usingthe Jones calculus and the Stokes parameters, since these are bothcomplete and commonly used. In terms of equation (1), the Jones vector Jthat describes the field at any location z or point in time t is givenby the following:

J=(A _(x)exp(iφ _(x)), A _(y)exp(iφ _(y)))=exp(iφ _(x))(A _(x) , A_(y)exp(−iδ)).

[0061] In practice, the factor exp(iφ_(x)) is ignored, so that the stateof polarization is described by the three main parameters: A_(x), A_(y)and δ. The physical interpretation of these three parameters is mostcommonly based on the polarization ellipse, which describes the pathtraced out by the tip of the electric field vector in time at aparticular location, or in space at a particular time. It should benoted that the Jones vector description is valid only for monochromaticlight, or a single frequency component of a signal.

[0062] A more complete description of the state of polarization is basedon the defined Stokes parameters, since this method also accounts forthe degree of polarization (DOP) of a non-monochromatic signal. In termsof the Jones vector parameters, the four Stokes parameters are definedby:

S ₀ =A _(x) ² +A _(y) ²

S ₁ =A _(x) ² −A _(y) ²

S ₂=2A _(x) A _(y) cos δ

S ₃=2A _(x) A _(y) sin δ,

[0063] and the degree of polarization (DOP), 0≦DOP≦1, is defined to be:${DOP} = {\frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}.}$

[0064] A partially polarized signal can be considered to be made up ofan unpolarized component and a polarized component. The DOP is used todefine that fraction of the signal which is polarized, and this fractionmay be described by either the polarization ellipse or Jones vector. Itis to be noted that, in strict terms, there are four parameters thatfully describe the elliptical signal: (1) the shape of the ellipse; (2)the size of the ellipse; (3) the orientation of the major axis; and (4)the sense of rotation of the ellipse. Thus, four measurements canunambiguously define the signal. These four parameters are often takento be A_(x), A_(y), the magnitude of δ, and the sign of δ. The fourStokes parameters also provide a complete description of fully as wellas partially polarized light. The Jones vector may be derived from theStokes parameters according to:

A _(x) ={square root}{square root over (S₀+S₁)}/{square root}{squareroot over (2 )}

A _(y) ={square root}{square root over (S₀−S₁)}/{square root}{squareroot over (2 )}

δ=arctan(S ₃ /S ₂)

[0065] It is to be noted that the last equation above does notunambiguously determine δ. Most numerical implementations of θ=arctan(x)define the resulting angle such that −π/2<θ<π/2. Thus, for S₂≧0, theexpression δ=arctan(S₃/S₂) should be used, where as for S₂<0, theexpression δ=arctan(S₃/S₂)±π should be used. Therefore, with theknowledge of the four Stokes parameters, it is possible to fullydetermine the properties of the polarized signal.

[0066] It has been recognized in accordance with the teachings of thepresent invention that the full state of polarization (SOP) cannot bedetermined by merely evaluating the signal passing through a singlepolarizer. Birefringence alone has also been found to be insufficient.In particular, a polarimeter may be based on a presumption that theoptical signal to be analyzed is passed through a compensator(birefringent) plate of relative phase difference Γ with its “fast” axisoriented at an angle C relative to the x axis (with the lightpropagating along the z direction). Further, it is presumed that thelight is subsequently passed through an analyzer with its transmittingaxis oriented at an angle A relative to the x axis. Then, it can beshown that the intensity I of the light reaching a detector disposedbehind the compensator and analyzer can be represented by:

I(A,C,Γ)=½{S ₀ +S ₁[cos(2C)cos(2[A−C])−sin(2C)sin(2[A−C])cos(Γ)]+S₂[sin(2C)cos(2[A−C])+cos(2C)sin(2[A−C])cos(Γ)]+S ₃ sin(2[A−C])sin(Γ)}.

[0067] In this case, S_(j) are the Stokes parameters of the lightincident on the compensator, such that S₀ is the incident intensity. Ifthe compensator is a quarter-wave plate (Γ=π/2), then the intensity asdefined above can be reduced to:

[0068]I(A,C,π/2)=½{S ₀ +[S ₁ cos(2C)+S ₂ sin(2C)]cos(2[A−C])+S ₃sin(2[A−B])},

[0069] Whereas if the compensator is removed altogether (Γ=0), theequation for the intensity I reduces to:

I(A,−,0)=½−{S ₀ +S ₁ cos(2A)+S ₂ sin(2A)}.

[0070] This latter relation illustrates conclusively that it isimpossible, without introducing birefringence, to determine the value ofS₃, and hence the sense of rotation of the polarization ellipse.

[0071] Following from the equations as outlined above, a polarimeter maybe formed using a compensator (for example, a quarter-wave plate), apolarizer, and a detector. In particular, the following fourmeasurements, used in conventional polarimeters, unambiguouslycharacterize the Stokes parameters:

[0072] 1) no wave plate; no polarizer→I(−, −, 0)=S₀

[0073] 2) no wave plate; linear polarizer along x axis→I(0,−,0)=½(S₀+S₁)

[0074] 3) no wave plate; linear polarizer at 45°→I(45,−,0)=½(S₀+S₂)

[0075] 4) quarter-wave plate at 0°; linear polarizer at45°→I(45,0,π2)=½(S₀+S₃).

[0076] In a conventional polarimeter using this set of equations, themeasurements may be performed sequentially with a single compensator,polarizer and detector. Alternatively, the measurements may be performedsimultaneously, using multiple components by splitting the incoming beamof light into four paths in a polarization-independent fashion.

[0077] Nonlinear Polarimeters:

[0078] Standard polarimeters measure the degree of polarization (DOP),or Stokes parameters that represent the polarization, by taking timeaveraged measurements of the x and y components of the E-field asrepresented by:

S ₁

=

E _(x) E _(x)

*

−

E _(y) E _(y)*

[0079] But, higher order moments can be measured as well as:

[0080]

E_(x)E_(x)*E_(x)E_(x)*

or

E_(y)E_(y)*E_(y)E_(y)*

[0081] A nonlinear polarimeter is a device that measures the higherorder moments. These measurements can provide extra information aboutthe bit stream or any polarized or partially polarized signal.

[0082] The number of moments that can be measured can be determined intwo ways. The E-field representation as mentioned above is one way:

[0083] Define (m,n) where m=# E_(x)'s and n=# E_(y)'s

[0084] This gives 1×(4,0)+1×(0,4)+2(1,3)+2(3,1)+3(2,2)=9 or

E_(x)E_(x)*E_(x)E_(x)*

E_(y)E_(y)*E_(y)E_(y)*

E_(x)E_(y)*E_(y)E_(y)*

E_(y)E_(x)*E_(y)E_(y)

E_(x)E_(x)*E_(y)E_(x)*

E_(x)E_(x)*E_(x)E_(y)*

E_(x)E_(y)*E_(x)E_(y)*

E_(y)E_(x)*E_(y)E_(x)*

E_(x)E_(x)*E_(y)E_(y)*

[0085] Alternatively, the un-averaged Stokes products S_(i)S_(j) can beconstructed. These are the 2^(nd) order moments before averaging:

S ₀ S ₁=(E _(x) E _(x) *+E _(y) E _(y)*)(E _(x) E _(x) *−E _(y) E _(y)*)

[0086] They are linear superpositions of the four product E fieldaverages. The independent quantities are:S₀S₁,S₀S₂,S₀S₃,S₁S₁,S₂S₂,S₃S₃,S₁S₂,S₂S₃S₃S₁. Again there are nine higherorder moments. Note that these are not the same as Stokes parameters:

S ₁ S ₂

≠

S ₁

S ₂

[0087] Also, S₀S₀ is not independent, because before averaging DOP=1,therefore, before time averaging, S₀S₀=S₁S₁+S₂S₂+S₃S₃.

What is claimed:
 1. A method of nonlinear polarimetry for measuringhigher order moments of the E field of light, comprising the steps of:receiving the light to be measured by a rotatable retarder; retardingthe light at a plurality of angles and at least two retardances;filtering the retarded light with a fixed polarizer; detecting thefiltered light with a linear and a nonlinear photodetector; calculatingthe spectra of the photodetector outputs to yield higher order momentsof the E field; and transforming the higher order moments to obtain thepolarization measurement.
 2. The method of claim 1 wherein detecting thelight from the analyzer with a linear and a nonlinear photodetectorcomprises detecting the light from the analyzer with a linear and anonlinear photodiode.
 3. The method of claim 1 wherein calculating thespectra of the detector outputs to yield the nine higher order momentsof the E field comprises calculating the Fourier spectra of the detectoroutputs to yield the nine higher order moments of the E field.
 4. Themethod of claim 1 wherein performing a transformation to obtain thepolarization measurement comprises performing a Stokes transformation toobtain the polarization measurement.
 5. The method of claim 4 whereinperforming a Stokes transformation to obtain the polarizationmeasurement comprises performing a Stokes transformation using the 9×9calibration matrix obtained from the nine higher order moments of the Efield to obtain the polarization measurement.
 6. The method of claim 1wherein detecting the light from the analyzer with a linear and anonlinear photodetector further comprises detecting the light from theanalyzer with a linear and a nonlinear photodetector and subtracting thelinear component from the nonlinear component.
 7. A static method ofnonlinear polarimetry for measuring higher order moments of the E field,comprising the steps of: receiving light to be measured in an opticalwaveguide with a plurality of waveguide birefringences; retarding thelight with the birefringences; scattering the light with polarizationsensitive gratings; detecting the scattered light from each polarizationsensitive grating with a photodetector to generate detector signals; andtransforming detector signals to polarization measurements.
 8. Themethod of claim 7 wherein receiving light to be measured in an opticalwaveguide with a plurality of waveguide birefringences comprisesreceiving light to be measured in an optical fiber waveguide with aplurality of fiber waveguide birefringences.
 9. The method of claim 7wherein receiving light to be measured in an optical fiber with aplurality of waveguide birefringences comprises receiving light to bemeasured in an optical planer waveguide with a plurality of planerwaveguide birefringences.
 10. The method of claim 7 wherein detectingthe scattered light from each polarization sensitive grating with aphotodetector comprises detecting the scattered light from eachpolarization sensitive grating with a photodiode.
 11. The method ofclaim 7 wherein detecting the scattered light from each polarizationsensitive grating with a photodetector further comprises detecting thescattered light from each polarization sensitive grating with aplurality of linear and nonlinear photodetectors.
 12. The method ofclaim 11 wherein detecting the scattered light from each polarizationsensitive grating with a plurality of linear and nonlinearphotodetectors further comprises detecting the scattered light from eachpolarization sensitive grating with a plurality of linear and nonlinearphotodetectors and subtracting the linear component from the nonlinearcomponent.
 13. A nonlinear polarimeter comprising: a retarder forreceiving light for polarization measurement and retard it at an anglewith a retardance, the retarder rotating; an analyzer optically coupledto the retarder, the analyzer fixed in place; and a photodetectoroptically coupled to the retarder to detect the optical signal from theanalyzer, the optical signal having a spectra related to the angularfrequency of the rotating retarder.
 14. The polarimeter of claim 13wherein the photodetector comprises a linear and a nonlinearphotodetector.
 15. The polarimeter of claim 14 wherein the photodetectorcomprises photodiodes.
 16. The polarimeter of claim 13 wherein theretarder has variable angle with retardance.
 17. The polarimeter ofclaim 13 wherein the retarder is fixed and the analyzer is a rotatinganalyzer.
 18. A nonlinear polarimeter comprising: an optical waveguidewith a plurality of birefringent sections to retard the light; aplurality of polarization sensitive gratings between the birefringentsections to scatter polarized light, wherein the polarization sensitivegratings and the birefringences are aligned according to a calibrationmatrix inverted, and a plurality of photodetectors, one photodetectoroptically coupled to the polarization sensitive gratings to receivelight from that grating, the photodetectors generating photodetectoroutputs, and the photodetector outputs can be transformed intopolarization parameters.
 19. The polarimeter of claim 18 wherein theoptical waveguide is a fiber waveguide.
 20. The polarimeter of claim 18wherein the optical waveguide is a planer waveguide.
 21. The polarimeterof claim 18 fabricated as an integrated optical structure.
 22. Thepolarimeter of claim 18 wherein there are nine polarization sensitivegratings and nine corresponding nonlinear detectors aligned such that a9×9 calibration matrix is invertible.
 23. The polarimeter of claim 18wherein each detector output is linearly related to a Stokes tensor. 24.The polarimeter of claim 18 wherein the photodetectors comprise linearand nonlinear photodetectors.
 25. The polarimeter of claim 19 whereinthere are 13 polarization sensitive gratings and nine correspondingnonlinear detectors and four linear detectors, the birefringences andgratings aligned such that a 13×13 calibration matrix is invertible. 26.The polarimeter of claim 18 wherein the photodetectors are photodiodes.